kingofthepirates wrote: ↑Sat Mar 30, 2024 9:07 pm
-1/12 is really wacky. I believe the process to get it is called Ramanujan summation. I'm learning calc rn, so I don't really know how/why it works (my teacher has expressed distaste towards the subject when a friend of mine brought it up), though the logic of the proof seems consistent.
JustAGuyNamedWill wrote: ↑Sat Mar 30, 2024 9:23 pm
It just
feels counterintuitive, even though the math (presumably) checks out.
Like, it should be impossible to add a positive number to another number and get less than what you started with, which is one.
CaptainFritz28 wrote: ↑Sun Mar 31, 2024 9:15 pm
I still refuse to believe that the sum of all natural numbers is -1/12, however. That just proves, to me, how limited our understanding of infinity is, because it would imply that -1/12 = infinity.
So, yeah, I am not a huge fan of people seeing "1 + 2 + 3 + 4 + 5 + ... = -1 / 12" before they have the mathematical background to put that statement in context, because it really is misleading, and can lead students learning calculus astray.. There are many other cool and surprising math results I would much rather talk about, but here we are.
In calculus (say, calculus 2 in a usual American education), the most important thing about series (infinite summations) is whether they converge (to a particular value) or diverge (which simply means don't converge). A series converges or diverges based on whether or not its corresponding sequence of partial sums converges or diverges. (To be extremely formal here, I should be including an epsilon-N definition of convergence of a sequence.) For example, the series
1/2 + 1/4 + 1/8 + 1/16 + ...
converges to 1, because the sequence of partial sums
1/2, 3/4, 7/8, 15/16, ...
converges to 1. As a non-example, the series
1 + 1 + 1 + 1 + 1 + ...
diverges, because the sequence of partial sums
1, 2, 3, 4, 5, ...
diverges to infinity. (Note: it is not allowed to say that a sequence "converges to infinity".) As a perhaps more interesting non-example, the so-called harmonic series
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ...
diverges, because the sequence of partial sums
1, 3/2, 11/6, 25/12, 137/60, ...
diverges to infinity (although this is certainly not obvious). So clearly, the series
1 + 2 + 3 + 4 + 5 + ...
diverges, because the underlying sequence of partial sums
1, 3, 6, 10, 15, ...
diverges to infinity.
So now that we have gotten that out of the way, what does it mean when some people say that
1 + 2 + 3 + 4 + 5 + ... = -1 / 12?
The answer is that some mathematicians have asked the question, "Yes, of course, this infinite series diverges. But if we had to assign a real value to it, what value would we assign to it?".
For example, consider the series
1 - 1 + 1 - 1 + 1 - 1 + ... .
It diverges, because the underlying sequence of partial sums
1, 0, 1, 0, 1, 0, 1, 0, ...
diverges. But consider for a moment the geometric series
x ^ 0 + x ^ 1 + x ^ 2 + x ^ 3 + x ^ 4 + ...
If |x| >= 1, then the series diverges, and if |x| < 1, then this series converges to 1 / ( 1 - x ):
x ^ 0 + x ^ 1 + x ^ 2 + x ^ 3 + x ^ 4 + ... = 1 / ( 1 - x ).
(As an example of that, if you set x = 1/2, then you obtain
1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 / ( 1 - 1/2 ) = 1 / ( 1 / 2 ) = 2,
which is almost exactly the same as the first result in this post (just add 1 to both sides).)
But what if you substitute in x = -1 into x ^ 0 + x ^ 1 + x ^ 2 + ... = 1 / ( 1 - x )? You obtain
1 - 1 + 1 - 1 + 1 - 1 + ... = 1 / ( 1 - (-1) ) = 1/2.
We just assigned a value to the divergent series 1 - 1 + 1 - 1 + 1 - 1 + ... . Now, that doesn't mean that it converges - it still very much diverges! And, of course, the answer feels weird: how can we add and subtract a bunch of integers and end up with something that is not an integer? But it is a value that we can assign to this divergent series. In a similar way to how we handled the geometric series, we can show that the formula
1 - 2 x + 3 x ^ 2 - 4 x ^ 3 + ... = 1 / ( 1 + x ) ^ 2
is valid for |x| < 1 (if you know calculus, just differentiate both sides of the geometric series formula above, and replace x with -x). Of course, once again, if |x| >= 1, the series diverges. But if you choose to ignore that, then substituting in x = 1 into the formula gives
1 - 2 + 3 - 4 + ... = 1 / ( 1 + 1 ) ^ 2 = 1 / 4,
which was previously mentioned in this thread. Again, it doesn't make sense - the series diverges! - but it is a way of assigning a value to this divergent series.
So then, the argument that 1 + 2 + 3 + 4 + 5 + ... = -1 / 12 is of a similar flavor. Of course the series 1 + 2 + 3 + 4 + 5 + ... is a divergent series. Of course it doesn't converge to a particular value. Of course, if you add up all of the natural numbers, the sum goes off to infinity. That's what I would hope that students would know and understand. But there exists a way to assign a value to this divergent series, and that value is -1 / 12.
There do exist applications in math of the statement "1 + 2 + 3 + 4 + 5 + ... = -1 / 12", but they are probably all at the graduate level or beyond. My preferred way to "prove" this statement isn't through the proof (mathematicians might prefer the term "heuristic") that Will posted, but rather by observing that, if you construct the analytic continuation of the Riemann zeta function zeta(s) to cover values of s with imaginary part <= 1, then zeta(-1) = -1 / 12, and zeta(-1) corresponds (in some sense) to "1 + 2 + 3 + 4 + 5 + ... ", but that is well outside the scope of this post.
Whew.